quantum field theory - What is the physical implication(s) of the isomorphism between ${rm SO}(2)$ and $mathbb{R}/mathbb{Z}$?

In the book Mathematical physics by V. Balakrishnan, he says (on page 329) that the isomorphism between ${\rm SO}(2)$ and $\mathbb{R}/\mathbb{Z}$, and the fact that $\mathbb{R}$ is the universal covering group of ${\rm SO}(2)$ has deep implications in two-dimensional systems in condensed matter physics and quantum field theory without any further explanation. Can someone elaborate on what he might have in mind?

Answer

In $d\ge 3$, the first homotopy group of $\mathrm{SO}^+(1,d)$ is $\mathbb Z_2$, which essentially leads to spin quantisation. In $d=2$, and due to $\mathrm{SO}(2)\sim\mathbb R/2\pi\mathbb Z$, we have $\pi_1(\mathrm{SO}^+(1,d))=\mathbb Z$, and therefore we don't have spin quantisation anymore. Particles are no longer classified into bosons vs. fermions, but they may have any statistics. We may find anyons, which lead to a very rich phenomenology (think fractional quantum Hall effect, etc.).

Recall that spin comes from the projective representations of the little group, to wit, $\mathrm{SO}(d)$. Unlike in higher dimensions, in $d=2$ we have that $\mathrm{Spin}(d)$ is not the universal cover of $\mathrm{SO}(d)$; indeed, $\widehat{\mathrm{SO}}(2)=\mathbb R$, which is non-compact. We thus no longer require $U(4\pi)=1$, so that spin is no longer a half-integer. Fun!